1. Write the converse, inverse, and contrapositive of. “If 1. Write the converse, inverse, and contrapositive of “If the measure of an angle is less than 90º, then the angle is acute.” ANSWER Converse: If an angle is acute, then the measure is less than 90º. Inverse: If the measure of an angle is not less than 90º, then the angle is not acute. Contrapositive: If an angle is not acute, then its measure is not less than 90º.

EXAMPLE 1 Use the Law of Detachment Use the Law of Detachment to make a valid conclusion in the true situation. If two segments have the same length, then they are congruent. You know that BC = XY. a. b. Mary goes to the movies every Friday and Saturday night. Today is Friday. SOLUTION Because BC = XY satisfies the hypothesis of a true conditional statement, the conclusion is also true. So, BC = XY. a.

EXAMPLE 1 Use the Law of Detachment b. First, identify the hypothesis and the conclusion of the first statement.The hypothesis is “If it is Friday or Saturday night,” and the conclusion is “then Mary goes to the movies.” “Today is Friday” satisfies the hypothesis of the conditional statement, so you can conclude that Mary will go to the movies tonight.

EXAMPLE 2 Use the Law of Syllogism If possible, use the Law of Syllogism to write a new conditional statement that follows from the pair of true statements. a. If Rick takes chemistry this year, then Jesse will be Rick’s lab partner.If Jesse is Rick’s lab partner, then Rick will get an A in chemistry. b. If x2 > 25, then x2 > 20. If x > 5, then x2 > 25. c. If a polygon is regular, then all angles in the interior of the polygon are congruent. If a polygon is regular, then all of its sides are congruent.

EXAMPLE 2 Use the Law of Syllogism SOLUTION The conclusion of the first statement is the hypothesis of the second statement, so you can write the following new statement. a. If Rick takes chemistry this year, then Rick will get an A in chemistry. b. Notice that the conclusion of the second statement is the hypothesis of the first statement, so you can write the following new statement. If x > 5, then x2 > 20.

EXAMPLE 2 Use the Law of Syllogism c. Neither statement’s conclusion is the same as the other statement’s hypothesis. You cannot use the Law of Syllogism to write a new conditional statement.

GUIDED PRACTICE for Examples 1 and 2 1. If 90° < m R < 180°, then R is obtuse. The measure of R is 155°. Using the Law of Detachment, what statement can you make? ANSWER R is obtuse

GUIDED PRACTICE for Examples 1 and 2 2. If Jenelle gets a job, then she can afford a car. If Jenelle can afford a car, then she will drive to school. Using the Law of Syllogism, what statement can you make ? ANSWER If Jenelle gets a job, then she will drive to school.

GUIDED PRACTICE for Examples 1 and 2 State the law of logic that is illustrated. 3. If you get an A or better on your math test, then you can go to the movies. If you go to the movies, then you can watch your favorite actor. If you get an A or better on your math test, then you can watch your favorite actor. ANSWER Law of Syllogism.

GUIDED PRACTICE for Examples 1 and 2 4. If x > 12, then x + 9 > 20. The value of x is 14 Therefore, x + 9 > 20 ANSWER Law of Detachment

EXAMPLE 3 Use inductive and deductive reasoning ALGEBRA What conclusion can you make about the product of an even integer and any other integer? SOLUTION STEP 1 Look: for a pattern in several examples. Use inductive reasoning to make a conjecture. (–2) (2) (–2) (–4) = –4, (–1) (2) = –2, = 4, 3 (2) = 6, = 8, (–1) (–4) 2 (–4) = –8, 3 (–4) = –12 2 (2) Conjecture: Even integer Any integer = Even integer

EXAMPLE 3 Use inductive and deductive reasoning STEP 2 Let: n and m each be any integer. Use deductive reasoning to show the conjecture is true. 2n is an even integer because any integer multiplied by 2 is even. 2nm represents the product of an even integer and any integer m. 2nm is the product of 2 and an integer nm. So, 2nm is an even integer. The product of an even integer and any integer is an even integer. ANSWER

EXAMPLE 4 Compare inductive and deductive reasoning Decide whether inductive or deductive reasoning is used to reach the conclusion. Explain your reasoning. a. Each time Monica kicks a ball up in the air, it returns to the ground. So the next time Monica kicks a ball up in the air, it will return to the ground. b. All reptiles are cold-blooded. Parrots are not cold-blooded. Sue’s pet parrot is not a reptile.

EXAMPLE 4 Reasoning from a graph SOLUTION Inductive reasoning, because a pattern is used to reach the conclusion. a. b. Deductive reasoning, because facts about animals and the laws of logic are used to reach the conclusion.

GUIDED PRACTICE for Examples 3 and 4 Use inductive reasoning to make a conjecture about the sum of a number and itself. Then use deductive reasoning to show the conjecture is true. 5. SOLUTION Conjecture: The sum of a number and itself is twice the number. Deductive reasoning: Let n be any integer. Use deductive reasoning to show the conjecture is true. n + n = 2n 

GUIDED PRACTICE for Examples 3 and 4 Give an example of when you used deductive reasoning in everyday life. 6. SOLUTION Check examples.

If water is at room temperature, then it is not frozen. ANSWER Daily Homework Quiz 1. Use the Law of Detachment to make a valid conclusion in the true situation. If two angles are complementary, then the sum of their measures is 90 . C and D are complementary. o m C + m D = 90 ANSWER o 2. Use the Law of Syllogism to make a valid conclusion in the true situation. If water is at room temperature, then it is a liquid. If water is a liquid, then it is not frozen. If water is at room temperature, then it is not frozen. ANSWER

Daily Homework Quiz Use the true statements below to determine whether the conclusion is true or false. Explain your reasoning. If Jeanine does her homework, then she goes to the movies. If Joaquin goes to the movies, then he buys popcorn. If Jeanine goes to the movies, then she buys popcorn. Jeanine does her homework. 3. Jeanine buys popcorn. ANSWER true; the Law of Syllogism 4. If Jeanine goes to the movies, Joaquin goes to the movies. ANSWER False; no laws of logic apply.